### Theory:

A set of numbers is said to be associative for a specific mathematical operation if the result obtained when changing grouping (parenthesizing) of the operands does not change the result.
Whole Numbers:
i) Addition: Changing the grouping of operands in addition to whole numbers, does not change the result. Hence, whole numbers under addition are associative.

ii) Subtraction: Changing the grouping of operands in the subtraction of whole numbers changes the result. Hence, whole numbers under subtraction are not associative.

iii) Multiplication: Changing the grouping of operands in the multiplication of whole numbers does not change the result. Hence, whole numbers under multiplication are associative.

iv) Division
: Changing the grouping of operands in the division of whole numbers changes the result. Hence, whole numbers under division are not associative.

Integers:
i) Addition: Changing the grouping of operands in addition to integers, does not change the result. Hence integers under addition are associative.

ii) Subtraction: Changing the grouping of operands in the subtraction of integers changes the result. Hence, integers under subtraction are not associative.

iii) Multiplication: Changing the grouping of operands in the multiplication of integers does not change the result. Hence, integers under multiplication are associative.

iv) Division: Changing the grouping of operands in the division of integers changes the result. Hence, integers under division are not associative.

Rational Numbers:
i) Addition: Changing the grouping of operands in addition to rational numbers, does not change the result. Hence, rational numbers under addition are associative.

$\left(\frac{a}{b}+\frac{c}{d}\right)+\frac{e}{f}=\frac{a}{b}+\left(\frac{c}{d}+\frac{e}{f}\right)$

ii) Subtraction: Changing the grouping of operands in the subtraction of rational numbers changes the result. Hence, rational numbers under subtraction are not associative.

$\left(\frac{a}{b}-\frac{c}{d}\right)-\frac{e}{f}=\frac{a}{b}-\left(\frac{c}{d}-\frac{e}{f}\right)$

$\left(\frac{2}{3}-\phantom{\rule{0.147em}{0ex}}\frac{3}{2}\phantom{\rule{0.147em}{0ex}}\right)-\phantom{\rule{0.147em}{0ex}}\frac{\left(-6\right)}{7}\ne \phantom{\rule{0.147em}{0ex}}\frac{2}{3}-\left(\phantom{\rule{0.147em}{0ex}}\frac{3}{2}\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}\frac{\left(-6\right)}{7}\right)\phantom{\rule{0.147em}{0ex}}$

iii) Multiplication: Changing the grouping of operands in the multiplication of rational numbers does not change the result. Hence, rational numbers under multiplication are associative.

$\left(\frac{a}{b}×\frac{c}{d}\right)×\frac{e}{f}=\frac{a}{b}×\left(\frac{c}{d}+\frac{e}{f}\right)$

$\left(\frac{2}{3}×\phantom{\rule{0.147em}{0ex}}\frac{3}{2}\phantom{\rule{0.147em}{0ex}}\right)×\phantom{\rule{0.147em}{0ex}}\frac{\left(-6\right)}{7}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\frac{2}{3}×\left(\phantom{\rule{0.147em}{0ex}}\frac{3}{2}\phantom{\rule{0.147em}{0ex}}×\phantom{\rule{0.147em}{0ex}}\frac{\left(-6\right)}{7}\right)\phantom{\rule{0.147em}{0ex}}$

iv) Division: Changing the grouping of operands in the division of rational numbers changes the result. Hence, rational numbers under division are not associative.

$\left(\frac{a}{b}÷\frac{c}{d}\right)÷\frac{e}{f}=\frac{a}{b}÷\left(\frac{c}{d}÷\frac{e}{f}\right)$

$\left(\frac{2}{3}÷\phantom{\rule{0.147em}{0ex}}\frac{3}{2}\phantom{\rule{0.147em}{0ex}}\right)÷\phantom{\rule{0.147em}{0ex}}\frac{\left(-6\right)}{7}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\frac{2}{3}÷\left(\phantom{\rule{0.147em}{0ex}}\frac{3}{2}\phantom{\rule{0.147em}{0ex}}÷\phantom{\rule{0.147em}{0ex}}\frac{\left(-6\right)}{7}\right)\phantom{\rule{0.147em}{0ex}}$